Topological Unified \((r, s)\)-Entropy of Continuous Maps on Quasi-Metric Spaces

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system’s guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	Dear authors! Submission of all materials is carried out only electronically through Online Submission System in personal account. DOI: 10.46698/p8176-1984-8872-z Topological Unified \((r, s)\)-Entropy of Continuous Maps on Quasi-Metric Spaces Kazemi, R. , Miri, M. R. , Mohtashami Borzadaran, G. R. Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 4. Abstract: The category of metric spaces is a subcategory of quasi-metric spaces. It is shown that the entropy of a map when symmetric properties is included is greater or equal to the entropy in the case that the symmetric property of the space is not considered. The topological entropy and Shannon entropy have similar properties such as nonnegativity, subadditivity and conditioning reduces entropy. In other words, topological entropy is supposed as the extension of classical entropy in dynamical systems. In the recent decade, different extensions of Shannon entropy have been introduced. One of them which generalizes many classical entropies is unified \((r,s)\)-entropy. In this paper, we extend the notion of unified \((r, s)\)-entropy for the continuous maps of a quasi-metric space via spanning and separated sets. Moreover, we survey unified \((r, s)\)-entropy of a map for two metric spaces that are associated with a given quasi-metric space and compare unified \((r, s)\)-entropy of a map of a given quasi-metric space and the maps of its associated metric spaces. Finally we define Tsallis topological entropy for the continuous map on a quasi-metric space via Bowen's definition and analyze some properties such as chain rule. Keywords: topological entropy, Tsallis entropy, Tsallis topological entropy, quasi-metric spaces Language: English Download the full text For citation: Kazemi, R., Miri, M. R. and Mohtashami Borzadaran, G. R. Topological Unified \((r, s)\)-Entropy of Continuous Maps on Quasi-Metric Spaces, Vladikavkaz Math. J., 2021, vol. 23, no. 4, pp. 56-67. DOI 10.46698/p8176-1984-8872-z + References 1. Shannon, C. E. A Mathematical Theory of Communication, ACM SIGMOBILE Mobile Computing and Communications Review, 2001, no. 1, pp. 3-55. DOI: 10.2307/3611062. 2. Renyi, A. On Measures of Entropy and Information, Hungarian Academy of Sciences, Budapest, Hungary, 1961. 3. Tsallis, C. Possible Generalization of Boltzmann-Gibbs Statistics, Journal of Statistical Physics, 1988, vol. 52, no. 1-2, pp. 479-487. DOI: 10.1007/BF01016429. 4. Rathie, P. N. and Taneja, I. J. Unified \((r, s)\)-Entropy and its Bivariate Measures, Information Sciences, 1991, vol. 54, no. 1-2, pp. 23-39. 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